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  "body": "\n## Summary\n\nA linear topology on a ring or a module is a topology in which the neighbourhoods of $0$ are generated by [[ideals|096Z9MP2ME]] or [[submodules|05WDSN597C]].\nIt makes the ring a topological ring whose small neighbourhoods of $0$ are closed under addition and under multiplication by ring elements.\nThe main examples are the [[I-adic topologies|2VF2TSJ3PY]], which lead to [[completions|KBN13NPYJQ]] such as the $p$-adic integers and formal power series rings.\n\n## Explanations\n\n### L1 human\n\nIn a linear topology, \"being close to $0$\" means \"lying in a small ideal\".\nTwo elements are close when their difference lies in a small ideal from a chosen family.\nFor the integers with the $p$-adic topology, the chosen ideals are $p^n \\Z$.\nSo $p^{10}$ is closer to $0$ than $p$, and the numbers $1, 1 + p, 1 + p + p^2, \\ldots$ get closer and closer to each other.\n",
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